One of the most striking theoretical consequences of General Relativity is the existence of wormholes and black holes.
While the second is currently investigated experimentally,
less is known about the first. Here we discussed something which in principle is very similar, quantum states which violate macro-locality.
Besides, it could be that the quantum state of the Universe is
a superposition of spacetimes with non-local links. In the present paper we considered such a possibility in a toy model constructed using the framework Quantum Graphity.
In order to do so, we had to extend the results of [1]} to a case in which the quantum state of the background is a superposition of many graph states.
The superposition of these graphs was chosen so that it is
dominated by a graph on which, as
we showed in earlier papers, the expectation values of number operators of the bosons hopping on it satisfy a closed equation for probability density in the classical
regime, i.e. a wave equation.
We extended the formula previously obtained and studied a particular case: graphs which violate micro- and macro- locality. As discussed, a violation of
macro-locality can be interpreted, within the model, as the presence of spatial non-local links in the background spacetime. This
is a concrete example of a quantum foam
within the framework of Quantum Graphity [2]}, [3]}.
The graph state was chosen on the basis of what we know from low energy physics, which is that Lorentz invariance is satisfied up and above the Planck scale[4]}.
We also used a class of graphs introduced in [1]}, rotationally invariant graphs. By exploiting their symmetry, the problem can be reduced to a 1-dimensional
one, i.e. Bose-Hubbard model on a line with specific couplings depending on the connectivity of the graph.
We thus constructed the states that are corrections to the low-energy physics by assuming that the non-local links are suppressed by a length
according to a certain
distribution. The length is measured by a combinatorial distance based on the low energy graph and which defines the state. We studied
for the cases \(d(x,y)=(x-y)^2\) 
and \(d(x,y)=|x-y|\) . We found that, in the continuum limit, there is no superluminal effect on the low-energy physics, i.e. the speed of propagation is intact. However,
there is an appearance of a mass dependence
on the constants of the distribution and that can be calculated within the model. These masses are square-positive and thus do not violate the physics
of the restricted Lorentz group, i.e.,
are not tachyonic. A simple analysis showed that this mass runs with the energy scale and, in particular, runs to zero at high energy. It is interesting to ask whether a
similar phenomenon happens
for the other fields. This analysis suggests the possibility that a quantum foam could contribute to the mass of a quantum field. As suggested in [6]} and [7]},
the possibility of having non-local link states within Loop Quantum Gravity is very natural. Also, it has been suggested that these states could contribute to the dark energy puzzle.
The results of the present paper suggests that, as in [7]}, the quantum foam contributes to the mass of fields hopping on such a superposition of spacetimes. We believe
that such possibility needs to be further investigated.
